An Approximate Coding-Rate Versus Minimum Distance Formula for Binary Codes

We devise an analytically simple as well as invertible approximate expression, which describes the relation between the minimum distance of a binary code and the corresponding maximum attainable code-rate. For example, for a rate-(1/4), length-256 binary code the best known bounds limit the attainable minimum distance to 65 ≤ d̃(n = 256, k = 64) ≤ 90, while our solution yields d(n = 256, k = 64) = 74.4. The proposed formula attains the approximation accuracy within the rounding error, and thus satisfies the condition of ⌊d⌋ ≤ d̃ ≤ ⌈d⌉, for ≈ 97% of (n, k) scenarios, where the exact value of the minimum distance d̃ is known. The results provided may be utilized for the analysis and design of efficient communication systems.